Recognition geometry I: maximum compatible matchings induce a pseudometric on uniform linear historiesles uniformes.

Resumen

The objective of this study was to demonstrate that the complement of the normalized similarity obtained through maximum compatible matchings defines a pseudometric on the space of uniform linear histories. These histories were conceived as finite directed acyclic graphs composed of disjoint unions of directed paths, with typed vertices and a common cardinality. Methodologically, basic research was conducted using a quantitative, theoretical, and deductive approach under a nonexperimental design. The procedure involved formally defining strictly realizable matchings subject to type preservation, injectivity, and order isomorphism between matched subposets. Their cardinality was then maximized to construct a similarity function, and recognition distance was defined as its complement. The results demonstrated that this distance satisfies nonnegativity, symmetry, zero self-distance, and the triangle inequality. The latter property was established through a fundamental subadditivity lemma based on the composition of maximum matchings and the inclusion-exclusion principle. Furthermore, the metric quotient called the Recognition Space was constructed, and its equivalence classes were shown to correspond exactly to the isomorphism classes of typed directed graphs. An illustrative example involving three histories of cardinality three confirmed the expected behavior of the calculated similarities and distances. It is concluded that the proposed combinatorial optimization problem generates a rigorous geometric structure for comparing uniform linear histories. This framework provides a mathematical foundation for future research on computational algorithms, spectral representations, functional embeddings, and the positive semidefiniteness of the recognition similarity function, thereby extending optimization-induced geometry to a new class of discrete combinatorial objects within contemporary mathematical and computational research.